i’m actually kind of reluctant to post this;
it’s probably the best idea i’ve had since
i started making up lectures without words
and now it’ll be easy to steal. you saw it
here first.

the Big triangle is made up of seven Little triangles.
each Little has seven Points. moreover the Points
of each Little (considered as subobjects of their Little)
are arranged in the same pattern
as the Littles themselves are (considered
as subobjects of Big).

pick any Little and call it L1.
there are three dark points in L1.
find the three Littles that correspond
to these three Points.

(for example, let L1 be the lower-left Little;
the three Littles i refer to now run along
the right side of Big [just as the dark Points
run along the right side of L1]).

the three Littles in question are then
precisely those in which the Point…
P1, say… that corresponds to L1
(considered as a subobject of Big)
is dark. i’ll be here all week.

James Tanton has a whole book of Math Without Words. His are all puzzles, and yours here is one of those theorems without words, I think. I don’t get it yet.

I see how a dark spot on one triangle corresponds to another triangle, which has a dark spot corresponding to the first triangle. mI see that it is allowed for a triangle to refer to itself that way (left and right middle do), and that each one has 3 dark spots.

How does one choose which spots go dark, and what does it have to do with the equation? Does that say P2(Z2) = P2(Z2)*? (I think you said somewhere that the star means dual, so I’m guessing it’s all 2’s, though the last looks like a 1.)

vlorbiksaid

P_2(F_2) is here a set of seven (so-called) points.
sometimes one calls ’em something like
[1], [2], … , [7] but i’ll just spare myself
the editing time and call ’em 1, … ,7.

[actually, any moment now i’m gonna want
the binary representation so i’ll go ahead
right now on second thought and admit that
these are themselves just nicknames for
001, 010, … ,111
(which are themselves “really” just nicknames
for the *official* names:
(0,0,1), (0,1,0), … (1,1,1)…
these objects will have started life
as the seven vectors connecting
the origin of a certain co-ordinate
system to a corner of its “unit cube”.
i’ve got some drawings but the margin
is too small to etcetera.)]

the eye-in-the-pyramid-lookin’ thing…
the “triangle pattern” that appears eight
times here… is then produced by arranging
these Points in such a way that seven Lines
are produced (these correspond to certain
planes through the origin in our “cube” model…
2-dimensional subspaces of F_2 (we’re “working
over” the-field-with-two-elements…
F_2 = Z_2 = { [0]_2 , [1]_2 }
if you just *have* to know [though really
one leaves off
the subscripts in actual calculations…
more “nicknames”…]) in other words.

the Lines of P_2 (associated with certain
planes of (F_2)^3 as i’ve just remarked)
can then be determined algebraically
using the “dot product” of vector analysis.

vlorbiksaid

but *really* they’re
L_001 = {010, 100, 100}
L_010 = {001, 100, 101}
…
L_111 = {011, 101, 110}.
the point here being that the Point
“abc” (with a, b, c \in {0,1} )
in on the Line L_ABC
(A, B, C \in {0,1}, natch)
precisely when the “dot product”
aA + bB + cC
is equal to zero (as an element
of F_2 of course).
see the previous post for some visuals.https://vlorblog.wordpress.com/2010/11/23/perhaps-this-will-refresh-your-memory/

the sets of Points and of Lines are the P_2(F_2) and P_2(F_2)*
of the title (of my drawing). abstractly, the “dual space” of
a “space” is the set of functions-to-the-base-field
(that preserve the algebraic structure); the dot-products
in question show that the dual space of the set-of-Points
can usefully be confused with the set-of-Lines (which itself
has the same “shape” as the set-of-Points itself…).

I’m wondering if you could start me out with context. This all sounds more interesting when I think about how the Platonic solids pair up into duals (the square has 6 faces and 8 vertices, the octagon has 8 faces and 6 vertices), with the tetrahedron having to act as its own dual.

vlorbiksaid

in P_2(F_2), “P_2” stands for “2-dimensional Projective space”
and “F_2” for “the Field with 2 elements”. makes this slightly clearer to some eyes.

so our context is “projective spaces”.
there are (on my very limited understanding)
two main ways to go about “constructing”
such spaces mentally.

i’ve sketched an outline of the “top-down” picture.
start with a vector space. , say
(the ordinary 3-D (x, y, z)-space of standard analysis…
“real three-dimensional space” as the saying has it.
now consider the set of lines through the origin
in this space. this set… with the “topology” derived
from our initial VS (“vector space”; R^3 in the example)…
is the projective space associated to our
“ordinary” space (remark: it has “dimension” *one less*
than that of the VS it’s associated to… the “lines”
in VS have collapsed to “points” of PS).

back in R^3. consider the unit sphere (S^3):
{ (x, y, z) \in R^3 | x^2 + y^2 + z^2 = 1 }.
any “line through the origin” of R^3 passes
through the sphere *twice*, in a pair ofantipodal points (like the north
and south poles [the best-known example]).
so P_2(R_2) is antipodal-pairs-on-the-globe.
some people including me find this easier
to “visualize” than lines-through-the-origin.
in particular, i can more clearly speak of
the “topology”… the question of “what points
are ‘close to’ what points”… since one can
draw little disks on the globe (antipodal
*pairs* of such disks, actually) to illustrate
(–literally!) the situation on a globe.
or on a piece of paper. P_2(R) can
be represented by “identifying points”
along the boundary of a square…
maybe you’ll have seen the “moebius
strip” drawn in this way (identify
a pair of opposite edges with
opposite orientations; on this model
one simply “sews a disc” along the
circle making up the boundary
of the moebius strip to produce
P_2(R)).

but enough of this hilarity.
on the “bottom-up” construction,
one begins with “ordinary” n-space
(over some field) and puts in *extra*
points (“ideal” points usually thought
of as “points at infinity”). more anon.
maybe.

anyhow, part of the fun here is that in projective spaces,
*any* two lines now meet at a point (perhaps “at infinity”),
just as *any* two points of (ordinary *or* projective) space
determine a unique line.

indeed, in P_2, a duality principle holds whereby
any true theorem *remains* true when “point”
is interchanged everywhere with “line” (mutadis
mutandis). amazing stuff really. projective space
turns out in some ways (“keep it simple”) to
be logically *prior* to euclidean space.

Hmm, the concept is so wacky at first… I think pictures that get at the conceptual stuff should come before pictures that get at the relationships among elements. (Am I right that the picture above does this?)

vlorbiksaid

each point “says” (in effect),
“the three lines-through-me
are here, here,
and here“. not that
one couldn’t already see that
already (where lines are *easy*
to see… but what about that
pesky *circle*?)…

typically in lower-division college maths,
F is the Real Number field and n=3;
this is the “ordinary 3-dimensional space”
of the contemporary (age of science)
worldview. x-y-z space; cogito ergo wow.
thingum again!

the so-called “dot product” of two vectors
in this context… which is defined (as we will
see**) as an *algebraic* object
[
which is to say, by applying the
*algebraic* operations built into
the so-called “field axioms”…
namely, additions(-and-subtractions),
and multiplications(-and-divisions)…
to certain “objects” (for instance,
to numbers or variables)
],
the “dot product” of two vectors, i say,
this *algebraically defined* gizmo,
turns out to be the crucial tool in
computing… and finally in talking about…
the *angle* formed by the two vectors
(and several closely-related *geometric*
properties of the situation at hand).

finally… oh sue v… or any other reader…
the relation to “duality” as i keep gesturing at
in blogetty-blogblogg-blogpost after BP
is along these lines.

v \dot w = 0

means (among other things)
that v and w (“vectors”)
are *at a right angle*
to each other.

one consequence of this is that
any line-through-the-origin
determines (“algebraically”!)
a unique plane-through-the-origin
(and vice-versa):

P= {w : w \dot v = 0} kinda thing.

and then you can turn right back around
and get the lines back from the planes.
that’s “duality”.

something very similar… some would
say “the *same* thing”… happens
again and again in situations where
“3-dimensional ‘real’ number space”
is replaced with “n-dimensional space
over an *arbitrary* number-field”:
every true statement about a given
r-dimensional subspace W
in this context will correspond
(“automatically”… which is to say,
because of some theorem-of-duality)
to a true statement about the
“dual space” of W (which is an
(n-r)-dimensional subspace of V).

it’s a “dot product” thing here, too
when you get down to it. which was
sort of my point.

**but. you know what.
i think i promised upthread to *define* that SOB.
but. to hell with it. are there no prisons.
are there no workhouses. should google
go broke just because they hate my browser.
look it up. thank me later.

V.

oh. ps. dammit.
there’s this bit where
n-dimensional *column* space (over F)
and
n-dimensional *row* space (over F)…
sue v and other teachers of “linear algebra”
will have already taken the point…
can be used to *generalize*
the (so-called) dot product.
(that’s why i made such a fuss about
V being a column-space… by habit
i wanted to be sure about the left-right
issues *just in case it came up*.)

the beginning of wisdom
is the study of finite-dimensional
vector spaces.

suevanhattumsaid

vlorbiksaid

let F be the field of seven elements:
F = (S, *, +), where S is some 7-set
({0,1,2,3,4,5,6}, say) and * and +
denote multiplication and addition
(“mod seven”).

F^2 is then a collection of 49 “points”
{(x,y): x \in F, y\in F}; it’s convenient
to think of this as “the (affine) *plane*
over F^2” and picture some such
array as
(0,0) (1,0) (2,0) … (6,0)
(1,0) (1,1)… (6,1)
…
(6,0)… (6,6).

the “lines” in this space are then
solution sets for linear equations
Ax+By=C.

suchlike “lines” can be visualized as
modifications of ordinary R^2 lines.
the relevant modifications are
(1) “throw out points having one or more
non-integer co-ordinates” and
(2)
“use the PacMan topology”
(one will recall, for example, that when
PacMan moves out of the right-hand
edge of his world, he comes back
at the corresponding spot on the
*left*-hand edge [and vice-versa];
likewise for top and bottom.
mod-7 arithmetic on the F^2 array
produces the same result visually
[essentially as a result of 6+1 =0…
this “wraparound” feature, by the way,
is why Rings are called “rings”]).

_ =____
=______
______=
_____=_
____=__
___=___
__=____,
with any luck, will appear as a sketch
of “x+y = 1”, for example.

anyhow, this collection of points-and-lines
is a pretty useful object in its own right…
but it’s not yet what we need for Spot It.

trouble is… while any two *points* determine
a unique *line*, it’s *not* true that any two *lines*
determine a unique *point* (instead, certain pairs
are “parallel” and have no point of intersection).

one can rectify this situation by “adding points”
to the geometry. specifically, one “adds in”
eight so-called “infinite points” (one for each
possible “slope” [the idea that “parallel lines
meet at infinity” is here made formal] and
one more infinite point for the “vertical” lines).

alas, the arithmetic gets slightly complicated.
the most convenient way to supplement our
array turns out to be to consider instead
certain collections-of-points of F^3.

the array considered earlier is, as it were,
“lifted up” into three space and place into
the plane z = 1. thus,
(0,0,1) (1,0,1)…(6,0,1)
(0,1,1) (1,1,1)…(6,1,1)
…
(0,6,1)… (6,6,1).
the “infinite points” are now
{(0,1,0), (1,1,0), (2,1,0)…(6,1,0)}\union
{(1,0,0)}.

drawing a picture might help to see that
these points are a complete set of representatives
of “lines through (0,0,0)” in F^3 (every such line
will contain exactly one of our 57 points).

it’s worthwhile to notice here that none of these
points is (0,0,0) and that the last non-zero
co-ordinate of each is 1. (the formalism is
that of “homogeneous co-ordinates”; this
need not concern us now.)

the lines of P^2(F)… the projective plane over
the field with seven elements are then defined
by fixing an element g \in G* and taking all the
points of G that “dot to zero” with g:
L(g) = {x \in G : g \dot x = 0}.

well, it turns out that this is exactly what we need
to get, not only “two points determine a unique line”,
but also “two lines determine a unique point”.
moreover, each *point* of G* is identified with
a *line* of G… but the structures G and G*
are identified by the *same* set-of-triples
(one has merely used [,] versus (,) for
extra clarity).

now we can make a Spot It deck.
create 57 easily-distinguished cartoons
and put ’em into one-one correspondence
with points of G.

make up 57 cards consisting of the “lines”
in G (as determined by the “dot product”
rule): seven eight cartoons on each card,
with each card-pair sharing exactly one
cartoon. throw out two cards, alas. done.

@sue v.
feel free to reprint in whole or in part;
i’d’ve run it myself in MMW if i hadn’t’ve
sworn off the prove-you’re-human
timesink frustrationware provided by
google. better still, i suppose, would be
to rewrite it in your own language for
your readers…

You tried to show me. I am a little bit happy (because I see there’s lots I’ll get to learn later). I don’t really get it, so can’t put it in my own words yet. I look forward to learning more about all this. Is there a text with lotsa problems you’d recommend? I think this might be a fun project for this summer (and I *will* be done working on my book by then).

vlorbiksaid

there’s a more-or-less standard university course
in geometry whose target audience typically consists
largely of future teachers; the thinking is that by considering
*other* axiom systems than the standard (euclidean plane),
students will come to a better understanding of what’s going
on *in* the euclidean context (and, with any luck, stop
thinking of *anything* as “obvious” until it’s been proven
and worked with for a while… this is the dreaded
“course in proving”. math majors should take some
version, too, but seldom do, i think).

anyhow, typically one considers the geometries
typically *called* “non-euclidean” (the parallel postulate
is here replaced either with “no parallels exist” or
with “infinitely many parallels to a line exist through
each point not on the line” [respectively “elliptic” and
“hyperbolic” geometry]) *and* (not “euclidean” but
also, somewhat weirdly, usually not *called* “non-
-euclidean”) others… projective geometries, for example.

the existence of small examples like the fano plane
(the photo of this post is a version) make PG’s rather
a charming subject for those of my bent (start with
the easiest interesting example and try to understand
it better and better). the *graphical* aspect is of course
another charming aspect of geometry (graphically
conceived; one is moving *away* from the pictures
typically [as the concepts sink in]…).

anyhow, such a class is where i saw first saw it;
i sat in (as an unofficial audit) in a version taught
by andrew lenard at IU in the 80’s.

then i taught a similar course in the early 90’s
when i was briefly a salaried pro.
i forget what text i used though.

bruce meserve _fundamental_concepts_of_geometry_
is a cheap dover reprint covering all this ground
masterfully. most instructors would probably reject
it as too mathy… contemporary textbooks typically
take a lot more pages to cover less ground.
i can’t put my hand on my copy just now.

the same could be said of dan pedoe’s
_geometry:_a_comprehensive_course_
(chapters vii & viii… but i haven’t studied
these nearly as carefully as the meserve).

i’ll go out on a limb here and conjecture that
howard eves _a_survey_of_geometry_ (1963)
(volume 1, chapter 6: projective geometry)
is something of a standard source.

that’s all that comes to hand. i’m moving so
my library’s a mess. there may be more in there
somewhere…. but of course the best thing to do
is to poke around in *your* (university’s) library.
check a bunch of indices; look the right stuff
in the right style; voila.

vlorbiksaid

but don’t let the rambling fool ya.
the set {(x,y,1)}\union{(x,1,0)}\union{(1,0,0)}
and the “dot product” operation are *all* we need
to generate a Spot It deck (in a “braindead” way…
forget the geometry and just *compute*).

each of our 7^2 + 7 + 1 (=57) objects is “dual”
to a set-of-8 (a “line”); these lines can be
computed with “brute force” (or a computer;
“python” is alleged to be well-suited for these
computations by those skilled in such matters).

i’ll retain the brackets for the dual space…
we can think of these points as “functions”…
but i’ll drop the commas and parens
(typing is hard).

(note that [of course!] these two lines share
a single point: 061 [the first to “spot it” wins]).

for L = L([231]), say?… more fiddling around is required,
but it’s the same game basically. there’s a “finite”
point on this line having x=0. let’s see: 3y+1=0;
3y= -1 = 6; y=2;
021\in L.
now for x=1. 2*1 + 3y +1 =0; 3y = -3; y=-1=6:
161 \in L.

so on. pictures help. it’s sort of relaxing if you can
get in the right groove… very like filling in crosswords.

and the *understanding* will come through the cracks
eventually… with any luck. so my best advice to you
if you want to see what i’ve been going on about
for these two years or so is to replace “7” with “3”
(any prime works) and make yourself a 13-card
Spot-It-deck analogue (with 4 cartoons to a card;
with 1 cartoon common to any card-pair).